Quantifying the Influence of the Crowded Cytoplasm on Small Molecule Diffusion

Abstract

Cytosolic crowding can influence the thermodynamics and kinetics of in vivo chemical reactions. Most significantly, proteins and nucleic acid crowders reduce the accessible volume fraction, ϕ, available to a diffusing substrate, thereby reducing its effective diffusion rate, D_eff, relative to its rate in bulk solution. However, D_eff can be further hindered or even enhanced, when long-range crowder/diffuser interactions are significant. To probe these effects, we numerically estimated D_effs for small, charged molecules in representative, cytosolic protein lattices up to 0.1 × 0.1 × 0.1 μm³ via the homogenized Smoluchowski electro-diffusion equation. We further validated our predictions against D_eff estimates from ϕ-dependent analytical relationships, such as the Maxwell-Garnett (MG) bound, as well as explicit solutions of the time-dependent electro-diffusion equation. We find that in typical, moderately crowded cell cytoplasm (ϕ ≈ 0.8), D_eff is primarily determined by ϕ; in other words, diverse protein shapes and heterogeneous distributions only modestly impact D_eff. However, electrostatic interactions between diffusers and crowders, particularly at low electrolyte ionic strengths, can substantially modulate D_eff. These findings help delineate the extent that cytoplasmic crowders influence small molecule diffusion, which ultimately may shape the efficiency and timing of intracellular signaling pathways. More generally, the quantitative agreement between computationally-expensive solutions of the time-dependent electro-diffusion equation and its comparatively cheaper homogenized form suggest that the latter is a broadly effective model for diffusion in wide-ranging, crowded biological media.

Graphical Abstract

Introduction

Intracellular biochemical reactions commonly rely on the diffusion of charged, small molecules¹ between regions where substrates are stored or synthesized to where they are ultimately utilized. These intracellular reactions can be strongly influenced by effective diffusion rates of their substrates,² which may be depressed by up to an order of magnitude relative to unrestricted diffusion in bulk solution.^3,4 In part, protein-, carbohydrate-, and nucleic acid-based ‘crowders’ restrict the intracellular volume accessible to diffusing substrates and thereby reduce their rates of transport (reviewed here⁵). However, additional factors beyond volume exclusion can equally influence the mobility of molecular diffusers. These factors include the distribution and activity of proteins or charged surfaces of organelles that selectively bind substrates, as well as the substrate’s shape and charge.^3,6-8 Improved quantitative predictions that delineate the relative contributions of these factors on shaping biomolecule diffusion is a challenging, but necessary, endeavor for understanding biochemical reactions and signaling in vitro and in vivo.^9-11

A wide range of experimental and simulation approaches have provided considerable insight into the effective diffusion rates of small molecules involved in intracellular signaling (see reviews^3,12). Methods such as fluorescence or raster image correlation spectroscopy,^13,14 and nuclear magnetic resonance^15-17 have yielded refined estimates for diffusion rates of small molecules like calcium, magnesium, and AMP in crowded intracellular media. A common theme emerging from these studies is that small molecule diffusion rates are substantially smaller than measurements in crowder-free solutions and are frequently anisotropic, owing to the organization of intracellular structures, like actin filament lattices. In part, these reductions in diffusion rates can be attributed to a reduced free volume fraction,³ although the nature of substrate/crowder interactions can substantially strongly modulate the effective rate.⁶ However, given the considerable variations in the sizes, molecular composition and electrostatic properties of cytoplasmic crowders, isolating the contributions of each factor via experimental means is challenging.⁹

In this regard, computational modeling is a strong complement to experimental techniques for describing the molecular underpinnings of experimentally observed diffusion rates and their influence on intracellular signaling.^9,18 A variety of computational approaches for modeling diffusion in crowded domains have been reported in the literature.¹⁸ The most common of which are based on explicit, particle-based representations of the diffuser or both diffuser and crowders, lattice-based models that assume discrete, finite-length hops for the diffuser, or spatially-continuous differential descriptions of the diffuser and its diffusion domain. These systems are prominently modeled as either deterministic or stochastic processes,¹⁸ or combinations thereof.¹⁹ Among these include particle-based approaches such as molecular dynamics^20-22 and Brownian dynamics^6,7,23-25 that explicitly represent the geometry of the diffuser and its crowders. Their computational complexity, however, generally limit their applications to nanometer spatial and microsecond temporal scales that are insufficient for describing sub-cellular transport. On the other hand, particle-based methods that represent diffusers as points, including Smoldyn²⁶ and MCell²⁷ scale well to longer timescales and to greater spatial extents appropriate for sub-cellular phenomena. Point representations, however, clearly sacrifice details of crowder shape and crowder/diffuser interactions that are known to influence transport and reaction kinetics.^18,24,28-31

As a compromise between these two extremes, partial differential equation (PDE)-based continuous diffusion models enable the preservation of atomistic details of crowder shape and composition, as well as non-bound interactions (like electrostatic forces³²) with an implicitly-represented diffuser.^33-35 Such PDE approaches are most appropriate for a continuum of diffusing species, like ions and nucleotides, that are fairly concentrated, significantly smaller than typical cellular crowders (100 kilodalton¹⁹) and do not substantially influence the local electrostatic potential of the medium. In earlier works, we have applied such continuum models to quantify the influence of molecular crowding around enzymes and electrostatic interactions on biochemical reaction rates and efficiency^29,30,36,37. Nevertheless, the computational expense grows exponentially with increasing number of explicitly-represented crowders, which renders these approaches impractical for the nanometer length scales relevant to sub-cellular transport phenomena.

To reduce this computational burden, multi-scale theories such as ‘homogenization’ can facilitate the extrapolation of molecular-scale information onto transport processes occurring over sub-micron and longer length scales.^38-41 In essence, homogenization theory posits that a continuum transport process occurs on two decoupled spatial scales: a microscopic scale within which the system is in steady-state, and a multi-fold larger macroscopic scale. This leads to a modified PDE (described in Sec. Methods) that is solved on the microscopic scale, which ultimately yields an effective diffusion tensor that is appropriate for transport on the macroscopic scale. We and others have shown that homogenized PDE models applied to ionic and nucleotide diffusion in structurally detailed, intracellular environments are in excellent agreement with experimentally measured trends in hindered diffusion rates and anisotropy.^8,42-48

To our knowledge, applications of homogenization theory to biomolecular systems has been limited to a single or a small number of crowder types, thus the challenge remains to understand transport in highly heterogeneous systems with highly variable protein sizes, positions, and charge distributions.⁵ In this study, we therefore consider the diffusion of a small, anionic molecule (such as adenosine monophosphate (AMP)) in an immobile, sub-micron scale lattice of common cytoplasmic proteins (Fig. 1), for which we precisely control the crowder distribution and charge density, as well as the electrolyte concentration. Specifically, we used realistic three-dimensional geometries of the cytoplasm, as well as two- dimensional (2D) and three-dimensional (3D) geometries comprised of charged, spherical inclusions to unravel molecular determinants of small molecule diffusion rates. Our modeling results for the realistic and 2D/3D simplified geometries are consistent with prior studies,^43,49,50 in that decreasing accessible volume progressively hinders diffusion, while long-range interactions can further slow or accelerate diffusion (repulsive versus attractive, respectively). Given that analogous trends are exhibited in both the 3D and 2D geometries, the majority of our simulations are done using the latter geometries, namely for rigorously examining the effects of ionic strength, surface charge potential and heterogeneous crowder shapes and distributions. Overall, our simulations offer quantitative insight into the effective diffusion rates of small biomolecules in heterogeneous cytoplasmic fractions, thereby providing important guidelines for electrokinetic transport in confined^51-53 or crowded media.

Figure 1:

Configurations of domains used in this study. a) Diffusion of charged particle (red spheres) in the absence of crowders (black spheres). Steady-state concentration gradient is indicated by higher concentration of diffusers (red background) at origin x=0 and lower concentrations (blue background) at x=L. b) Same as a), but with neutrally-charged diffuser (black spheres) c) Same as b), except crowders are assigned repulsive electrostatic potentials.

Methods

Structural models of the crowded cytoplasm

The model cytoplasm considered in this study was based on Brownian dynamics (BD) trajectory snapshots of common cytosolic proteins and nucleic acids in E. coli by McGuffee et al²⁴ (see Fig. 2). We assume lattice proteins are immobile relative to the rapidly diffusing small molecule, given the large difference in diffusion constants (e.g. diffusion constant for AMP is at least ten-fold greater than the median for E. coli cytoplasmic proteins⁵⁴). Our numerical approach (see Sec. Volumetric mesh generation for cytoplasm model) uses three-dimensional finite element meshes based on these BD data, which were constructed by 1) exporting their solvent-accessible molecular surfaces from VMD,⁵⁵ 2) meshlab⁵⁶ refinement to remove defects 3) tetrahedralization via tetgen⁵⁷ 4) conversion to .xml format for compatibility with the PDE solver FEniCS⁵⁸ via dolfin-convert. Details of this procedure are available in the Supplement (Sec. Volumetric mesh generation for cytoplasm model). The final result was a finite element mesh of the representative volume element (RVE) representing the aqueous phase between the crowders, of which an example is shown in Fig. 4a.

Figure 2:

a) 808 x 808 x 808 Å snapshot of crowded bacterial cytoplasm simulation from McGuffee et al.²⁴ b) A prototypical representative volume element (RVE) used for homogenization is represented in color.

Figure 4:

a) Three-dimensional solution of homogenized diffusion equation for an uncharged diffuser (see Eq. 4) based on the Ångstrom-resolution cytoplasm data in Fig. 2. b) HSE-predicted effective diffusion constants along the x (red), y (yellow) and z (black) directions as a function of RVE size (100, 200 and 300 Å) and the quantitative comparison of them with the analytical Hashin-Shtrikman (HS) upper bound.

We additionally created lattices of cylindrical (2D) or spherical (3D) primitives via gmsh⁵⁷ to afford more control over testing the influence of crowder shape, size, and surface charge on diffusion rates. The size distribution were chosen to best approximate those of the original cytoplasm model, based on our characterization of the crowder radius of gyration (R_g) (see Fig. S2), for which the average R_g was approximately 30 Å. The appropriate accessible volume fraction (accessible to the diffusing species), ϕ, was determined based on simulation data from McGuffee et al²⁴ that used 1008 proteins within an 808 x 808 x 808 ų box, or ϕ = 0.78. We note that our definition of ϕ as the accessible volume fraction is consistent with⁵⁹ and our earlier publications,^8,45 although in other contexts the volume fraction is based on the inaccessible fraction (e.g. 1 − ϕ).

Based on a perfect simple cubic arrangement of R_g=30 Å crowders with a center of mass distance, d_com, of 80.6 Å would yield ϕ = 0.78, which is consistent with crowding estimates in E. coli.⁴ Given the computational expense in evaluating large numbers of three-dimensional finite element meshes, we additionally defined two-dimensional reference systems using d_com=80.6 Å (based on surface distances for spherical crowders) and d_com=113.0 Å for volume fractions of ϕ = 0.57 and ϕ = 0.78, respectively. We consider both volume fractions in order to examine the potential influences of inter-crowder distances and crowder volume fractions analogous to the 3D system. The inter-crowder distance, H, is related to the distance between centers of mass (e.g. d_COM = H + 2 · R_g); while d_com may be a more familiar metric, H is better suited for assessing the range of electrostatic interactions between charged protein surfaces. Where indicated in the text, we additionally randomize the protein size and position in accordance with Fig. S2, based on a Monte Carlo protocol (Sec. Monte Carlo protocol for generating crowder ensembles). We additionally assumed crowders have boundary potentials of ψ₀ = ±19.2 [mV], based on ζ potential measurements ranging from 0.5 to 0.75 k_BT for bovine serum albumin in 150 mM potassium chloride (KCl) and approximately neutral pH.⁶⁰ These parameters are summarized in Table S1.

Diffusion models

Our numerical approach is based on a time-dependent diffusion model governing the diffuser concentration c(x, t) on a domain Ω_m,

$$\frac{\partial c}{\partial t}=-\nabla \cdot D\nabla c\text{on}{\Omega }_m,$$ (1)

where D is the small molecule diffusion constant. To reflect the influence of an electrostatic potential of mean force acting on the diffuser, we formulate the concentration function as a Smoluchowski equation such that

$$\frac{\partial c}{\partial t}=-\nabla \cdot D{e}^{-\beta q\psi }\nabla (e^{\beta q\psi }c)\text{on}{\Omega }_m,$$ (2)

$$=-\nabla \cdot \tilde{D}\nabla \tilde{c}\text{on}{\Omega }_m,$$ (3)

where q is the diffuser charge, ψ is the electrostatic potential, β ≔ 1/k_BT with k_B the Boltzmann constant, and T the temperature. The second line reflects the Slotboom form using $\tilde{D}(y)≔D(y)e^{-\beta q\psi (y)}$ and $\tilde{c}(y)≔e^{\beta q\psi (y)}c(y)$. The evaluation of the electrostatic potential arising from charged crowders is described in Sec. Poisson-Boltzmann model for the electrostatic potential.

Direct numerical simulation of Eq. 2 can yield effective diffusion parameters, as we describe in Sec. Estimating effective diffusion rate from continuum diffusion simulations; in practice, however, this entails evaluating the PDE over a considerable number of time intervals, which can be prohibitively expensive. Therefore, we propose solving a ‘homogenized’ steady-state problem that directly yields effective diffusion constants, albeit at a computational cost that can be at least an order of magnitude smaller. The two-scale homogenization approach^39,61 applied to Eq. 2 assumes that Ω_m has two length scales of interest, x and y ≔ x/ϵ where the constant ϵ is assumed to be small compared to the dimensions of Ω_m and that Ω_m is periodic in y where y corresponds to a nanometer length-scale, whereas x corresponds to sub-micron or longer. The y-scale periodic unit, otherwise known as the RVE, is denoted by Ω, within which the accessible region is defined as Ω_ϵ. This relationship is illustrated in Fig. 2b), where the RVE Ω (colored) is a subset of the macroscopic domain Ω_m (gray). We further assume that D is y-periodic, such that D(x, y) can be written as D^ϵ(y).

Before continuing, we briefly discuss the implications of this assumption. Namely, assuming ‘microscopic’ periodicity allows us to represent the influence of interacting crowders on diffusion implicitly through an effective diffusion coefficient, D_eff. In practice, time- and spatially-dependent simulations of intracellular signaling in cell-shaped geometries could then proceed using the D_eff in Eq. 2, for which the D_eff term effectively ‘coarse-grains’ the impact of crowders on sub-cellular diffusion.

Two-scale homogenization proceeds by rewriting Eq. 2 using the expansion of c in powers of ϵ (i.e. $c={\sum }_i{c}_i{ϵ}^i$) and the revised gradient operator (i.e. ∇ = ∂_x + ϵ⁻¹∂_y). In previous work,^8,45 we demonstrate that the resulting equation yields the homogeneous y-scale steady state problem: find χ such that

$$\begin{gathered} \nabla \cdot \left(D_{ij}^{ϵ}({\delta }_{jk}+\frac{\partial {\chi }_k}{\partial y_j})\right)=0,\text{on}{\Omega }_{ϵ}, \\ \left(D_{ij}^{ϵ}({\delta }_{jk}+\frac{\partial {\chi }_k}{\partial y_j})\right)\cdot \widehat{n}=0,\text{on}{\Gamma }_{ϵ}≔\partial {\Omega }_{ϵ}, \\ \left(D_{ij}^{ϵ}(\frac{\partial {\chi }_k}{\partial y_j})\right)\cdot \widehat{n}=0,\text{on}\Gamma \setminus {\Gamma }_{ϵ}≔\partial \Omega \setminus \partial {\Omega }_{ϵ}, \end{gathered}$$ (4)

where Γ_ϵ corresponds to a molecular boundary and Γ\Γ_ϵ is typically the RVE boundary, while D^ϵ is the small molecule diffusion rate within the RVE. Eq. 4 is similar to Eq. 2, but includes the Kronecker delta, δ_jk, which essentially couples the microscopic and macroscopic coordinate systems. Given solutions for χ, the effective diffusion tensor (applicable to Eq. 2 on the macroscopic domain) is determined via

$$D_{ij}=\frac{1}{\mid \Omega \mid }{\int }_{{\Omega }_{ϵ}}{D}_{ij}^{ϵ}(y)\left({\delta }_{jk}-\frac{\partial {\chi }_k}{\partial y_j}\right)dy$$ (5)

where Ω is the RVE volume. In this study, we report normalized self-diffusion constants along the x-direction, e.g. ${\mathrm{D}}_{\text{eff}}\equiv \frac{D_{xx}}{D_{xx}^{ϵ}(y)}$, except where otherwise noted. The reader may consult ^8,42,61,62 for additional details of the derivation. The PDEs defined in this study were numerically evaluated via the finite element method using FEniCS,⁵⁸ for which a piece-wise linear basis and the default direct linear solver were used. Details of the numerical procedure are identical to protocols outlined in.^8,45 All code written in support of this publication will be publicly available at https://bitbucket.org/pkhlab/pkh-lab-analyses. Simulation input files and generated data are available upon request.

Results and Discussion

Diffusion of a neutral diffuser in the crowded cytoplasm

Model validation for two- and three-dimensional domains

The homogenized Smoluchowski equation (HSE) enables the prediction of effective diffusion parameters based on the crowder configurations in nano- to micron-scale domains. We first demonstrate the accuracy of our implementation for a neutral diffuser in the two- and three-dimensional crowder domains in Fig. 3a-b by 1) approximating the effective diffusion constant using semi-analytical bounds and 2) explicitly solving the time-dependent diffusion equation (Eq. 1) in the presence of crowders to obtain an effective diffusion constant. Fig. 3c depicts the normalized effective diffusion coefficients of a neutral diffuser predicted via HSE (solid symbols) for two- and three-dimensional crowder domains with accessible volume fractions ranging from ϕ = 0.5 – 1.0. For ϕ → 1.0, D_eff approaches to the (normalized) bulk rate at small occluded volume fractions. As the free volume fraction decreases toward ϕ = 0.5 with increasing crowder density, D_eff monotonically decreases to roughly 30% and 40% of its bulk value for spherical and cylindrical inclusions, respectively. We further note that D_effs predicted for cylindrical inclusions tend to be slightly smaller (up to approximately 25%) than those for spherical crowders; therefore our simulations using two-dimensional geometries will very modestly overestimate the effects of crowders on small molecular diffusion in analogous three-dimensional domains. Nevertheless, near typical intracellular volume fractions (ϕ ≈ 0.78), D_effs for both spherical and cylindrical inclusions are comparable at roughly approximately 60-70% of the bulk diffusion rates. These HSE predictions are consistent with estimates based on two semi-analytical bounds for uniformly distributed inclusions, the Hashin-Shtrikman (HS) upper bound⁶³ for spherical crowders (solid line),

$$D_{HS}=\frac{2\varphi }{(3-\varphi )}$$ (6)

and the MG bound for cylindrical particles (dashed line),⁶⁴

$$D_{MG}=\frac{\varphi }{(2-\varphi )}$$ (7)

where ϕ is the accessible volume fraction. Lastly, we find comparable D_eff predictions based on explicit time-dependent simulations of the continuum diffusion equation (open symbols, see Sec. D_eff from continuum diffusion simulations for more information and error analyses). Given that D_eff estimates for two- and three-dimensional lattices show similar dependencies on accessible volume fraction and are comparable in magnitude, subsequent analyses will primarily use 2D geometries for reasons of computational efficiency, except where otherwise indicated.

Figure 3:

Two- (a) and three-dimensional (b) simulations domains used for validating continuum and homogenized Smoluchowski equation (HSE) models of hindered diffusion. c) Comparison of 2D (red) and 3D (black) effective diffusion coefficient for HSE (solid symbol) and continuum model (open symbol) as a function of accessible volume fraction with appropriate analytical bounds for non-interacting crowders (Maxwell-Garnett (MG) for cylinders and Hashin-Shtrikman (HS) for spheres) are shown with dashed and solid lines, respectively.

Influence of protein shape and distribution on effective diffusion rates for neutral diffusers

After validating our HSE model against primitive, uniformly distributed crowders, we next determined D_eff based on three-dimensional snapshots from the McGuffee’s BD simulation data. We note that the HSE model permits a computationally inexpensive evaluation of D_eff for a complex, crowder-laden cytoplasmic domain (several minutes on 16 processors), and to our knowledge, is the first application of homogenization to a structurally-detailed, Ångstrom-resolution cytosol fraction. In Fig. 4a, we present the three-dimensional solution for ξ (see Eq. 2); generally speaking, the reddish and bluish tones represent regions where diffusion is more strongly hindered relative to the grayish regions. In Fig. 4b we report the effective diffusion constants along the x, y, z axes for RVE sizes of 100-300 Å. The diffusion tensors are nearly isotropic, with ranges from 0.40 and 0.55; such a result is not surprising, given that the cytoplasmic proteins are roughly spherical in shape and have no preferential, anisotropic arrangement. Furthermore, our predictions are consistent with estimates indicating that translational diffusion rates are within an order of magnitude of bulk diffusion rates for diffuser sizes < 500 kilodalton^3,4 and closely match the D_eff=0.51 predicted by the HS bound. The D_eff predictions from these data have no obvious dependence on the RVE size, which we attribute to small sample sizes (one case for each cell length).

To better understand the variance of effective diffusivity with respect to crowder positions and sizes, we characterized the R_g (see Fig. S2) of crowders reflected in the McGuffee cytoplasm data. The skewed distribution had an average R_g of approximately 30 Å, with prominent peaks at 15 and 30 Å, although some radii were as much as 80 Å in size. Given the sensitivity of the effective diffusion tensor to the free volume fraction and the difficulty in constructing atomistic-resolution three dimensional meshes of complex topologies,⁶⁵ we generated randomized, 2D crowder distributions (11 per cell length ranging from 100-2000 Å). Specifically, each randomized distribution was chosen such that their radii of gyration well-approximated the statistics derived from the BD data shown in Fig. S2. Their randomized positions were generated using a Monte Carlo approach (Sec. Monte Carlo protocol for generating crowder ensembles) that ensured crowders were non-overlapping and within the RVE boundaries. We note that the accessible volume fraction of the original three-dimensional geometry could be modulated by shrinking or enlarging the solvent probe radius for generation of the Connolly solvent-accessible surface.⁶⁶ However, there is considerable burden of manually revising commonly-occurring meshing defects⁶⁵ such as overlapping and self-intersecting vertices. Moreover, our Monte Carlo approach permits more exhaustive sampling of crowder locations and spatial configurations that would be possible from the original Brownian dynamics trajectory data made available by McGuffee et al.²⁴ In Fig. 5, we report the mean D_effs as a function of RVE size for configurations with randomized positions and radii (RandPos/RandRad) as well as randomized positions and constant (R_g=30Å) radii (RandPos/ConstRad). We found that D_eff≈0.65 across RVE lengths, regardless of whether randomized or fixed radii were used. As expected for larger sample sizes, the standard error in the mean D_eff decreased with increasing RVE size. These data therefore indicate that for neutral diffusers, 1) reliable D_eff estimates may be obtained using relatively small lengths (300 nm) provided sufficiently large sample sizes and 2) D_eff estimates are apparently insensitive to crowder position and size variations at cytosolic volume fractions (ψ ≈ 0.78). In other words, fine-resolution details of cytosolic globular proteins’ three-dimensional shape are likely unnecessary for predicting their influence on diffusion rates of small molecules. Therefore, we expect that the deviations in D_eff as a function of direction and cell size relative to the HS bound in Fig. 4 were due to small sample sizes, as opposed to variations in shape. We clearly anticipate exceptions to this trend if the protein positions or shapes are strongly anisotropic, such as for myosin/actin filaments in contractile cell,⁶⁷ especially if the proteins strongly interact with the diffuser (examined in the next section).

Figure 5:

Predicted effective diffusion rates, D_effs, for a neutral diffuser in crowded, two-dimensional RVEs with volume fraction ϕ = 0.78, as a function of domain size (cell length). Geometries are derived from the crowder distributions in Fig. 2.

Diffusion of a charged molecule in the crowded cytoplasm

Effects of protein charge on electric potential

The electrostatic potential arising from surface-exposed polar amino acids is known to influence protein-protein and protein-ligand association thermodynamics and kinetics.^68-70 Therefore, it can be expected that the electrostatic interactions of a small, charged diffuser with densely-packed, immobile proteins comprising the cytoplasm would influence their effective diffusion rate, although the extent to which 1) typical protein surface potentials and 2) protein distribution in large sub-micron cytoplasmic fractions has not been well-explored. Before investigating these contributions, we first predict the electrostatic potential from distributions of charged 2D crowders by numerical solution of the Poisson-Boltzmann (PB) equation. Further details on the determination of the electrostatic potential and assignment of crowder surface potential can be found in Sec. Estimating effective diffusion rate from continuum diffusion simulations.

In Fig. 6, we predict the electric potentials arising from a uniform distribution (ϕ = 0.78) of crowders with −19.2 mV surface potentials subject to ionic strengths ranging from 5 to 500 mM. The 2D surface plot electric potential distributions are shown, as well as the spatial distribution of electric potential between two neighboring particles in one dimension (see Fig. 6b). The ionic strength is reported as the dimensionless quantity, κH, for which κ is the inverse Debye length, and H is the edge-to-edge average distance between proteins.

Figure 6:

Electric potential in (a) two dimensions and (b) one dimension between negatively-charged crowders for various background KCl concentrations (5, 20, 150, and 500 mM, corresponding to κH of 1.06, 2.14, 5.86, and 10.69 respectively), given crowder edge spacing (H) of 5.2 nm and ϵ=−0.75 k_BT.

The magnitude of the electrostatic potential is clearly maximal at the protein surface (ψ₀=−19.2 [mV]) and minimal midway between neighboring proteins. As would be expected for strong electrolyte solutions, the decay of the potential as a function of distance from the crowder increases with increasing ionic strength. At κH = 10.69 corresponding to a concentration of 500 mM, the electric potential rapidly decays to zero within 4.3 Å of the crowders, thus the influence of electrostatic interactions are expected to be negligible within most of the accessible diffusion volume. However, as κH decreases, significant overlap of potentials from neighboring crowders is observed throughout the entire domain. To understand the impact of this potential on the distribution of charged diffusers, we assume local equilibrium inside the RVE, which is commonly done to ensure validity of the constitutive model at the microscale despite non-equilibrium conditions at the macroscale.⁴⁰ In this regime, the Boltzmann relation c(x, z_ieψ) = c₀exp [−βz_ieψ] may be used to describe the diffuser concentration, c, in the diffuse layer (DL) relative to the bulk solution (c₀). Hence, electronegative DLs are predicted to exclude negatively-charged diffusers, which is analogous to increasing the crowders’ effective radii; in complement, attractive forces would localize positively-charged diffusers toward the charger, thereby reducing the crowders’ effective size. This interpretation holds for the weakly attractive electrostatic interaction energies assumed in this study (less than −1 to −2 k_BT) and low diffuser concentrations relative to the bulk electrolyte; beyond this regime, the diffuser would be expected to compete with the bulk electrolyte and potentially decrease the Debye length.

To better approximate the diverse electrostatic properties of proteins comprising the cytoplasm, we also present electric potentials corresponding to an electro-neutral system (nearly equal numbers of positively- and negatively-charged crowders) for alternating (Fig. S4) and randomly-distributed crowders (Fig. S5). For alternating charged crowders, ψ = 0 midway between crowders (e.g. d=H/2). Despite the inverse symmetry of the electric potential across the midpoint, however, an asymmetric distribution of diffusers is expected, as c(r = 0, z_ieψ = 0.75kT) ≈ 0.47 and c(r = 0, z_ieψ = 0.75kT) ≈ 2.12 for diffusers with z_i = 1 and z_i = −1, respectively. For randomly distributed charged crowders (Fig. S5), we observe localized electropositive (red) and electronegative (blue) regions that we expect will give rise to strongly heterogeneous diffuser distributions.

Effective diffusion rates as a function of charged crowder distribution

To investigate the influence of charged crowders on small molecule diffusion, we evaluated the HSE subject to the PB solutions from the previous section. Similar to the neutral diffuser case in the previous section, we predicted D_effs for charged cylindrical and spherical crowders; however, since we were unaware of effective diffusion rate relationships that account for both electrostatic interaction strength and volume fractions, we validated our HSE predictions against the time-dependent Smoluchowski model (Eq. 2). Namely, in Fig. 7 we compute the normalized effective diffusion coefficient for charged diffusers (z=−1, 0, +1) using a lattice of cylindrical crowders with varying volume fractions (ϕ = 0.5 – 1.0). For these cases, we assumed that the crowders have uniform, negatively-charged surface potentials of −19.2 mV and are immersed in a background 150 mM KCl solution (λ_D = 7.8Å).

Figure 7:

Predicted normalized diffusion coefficients based on negatively-charged crowders with volume fraction ϕ, similar to the configuration in Fig. 6. Coefficients are computed using the HSE (open symbols), continuum (closed symbol-dashed line), Maxwell-Garnett (MG) bound (cross-symbol). Diffuser charges of z=−1 (circle), z=0 (square), and z=+1 (triangle) are given, assuming C_KCl=150 mM and ψ₀ = −19.2 mV (0.75k_BT for q=1) for the crowders. The black dotted-line corresponds to the center of mass distance between crowders. We present an analogous plot for spherical inclusions in Fig. S6.

Similar to our results assuming a neutral diffuser, the normalized D_eff decreases with reduced accessible volume fraction regardless of charge, although the rates of decrease are charge-dependent. While the effective diffusion rates of the neutral diffusers (z=0, black squares in Fig. 7) follow the MG analytical bound (red crosses), negatively charged diffusers exhibit slower diffusion (blue dots). In contrast, for attractive interactions (red triangles), the normalized diffusion coefficients exceed those of the uncharged and negatively charged diffusers. Moreover, the HSE results (hollow symbols) and the time-dependent continuum models (line-symbols) are in quantitative agreement, which validates our use of the HSE in modeling electrokinetic phenomena, albeit at a fraction of the computational expense. We attribute hindered diffusion under repulsive interactions to the diffuser’s exclusion from the crowders’ DLs, as illustrated in Fig. 6. In contrast, the diffusional acceleration by weakly-attractive interactions observed here and in related experiments^{7,23,29,30,43,71-73} has been rationalized as the ‘smoothing out’ of the chemical potential energy surface roughened by crowder-induced excluded-volume regions.⁷

We additionally examined the dependence of charged-mediated D_effs on the electrostatic interaction strength. In Fig. 8a, D_effs are reported from solutions based on the time-dependent diffusion equation, where ψ is determined from the nonlinear (solid line) and linearized (dashed-line) PB equation, and based on the HSE (squares symbols) using the linearized PB equation. We assumed for these calculations a physiologically-reasonable volume fraction (ϕ = 0.78) and a bulk electrolyte with λ_D = 7.8Å. The predicted D_effs from these three approaches are in quantitative agreement, with the most hindered diffusion rates (0.54) reported at ψ = −25.6 mV, corresponding to an electrostatic interaction energy of ϵ ≡ qψ = 1 k_BT, given z=−1 for q ≡ ze; on the other hand, the fastest rates (0.81) were found for ϵ → −1 k_BT. In the absence of electrostatic interactions, ϵ = 0 k_BT, and we recovered the MG bound (D_eff=0.64).

Figure 8:

Left) Normalized effective diffusion coefficient of the diffuser as a function of electrostatic potential energy, ϵ = qψ [k_BT], where we have assumed crowder surface potentials of ∣ψ∣ < 25.6 [mV] and z=−1 for q ≡ ze. Results based on the linearized and nonlinear PB equations are compared with HSE when λ = 7.85 Å which corresponds 150 mM salt concentration and ϕ = 0.78. Right) Logarithm of D_effs from left figure is shown, in addition to a linear fit to D_effs predicted at 1 mM ionic strength, based on Eq. 8.

It is interesting to note that log(D_eff) is nearly linear with respect to the electrostatic interaction energy as shown in Fig. 8b for $\log \frac{D(\psi )}{D_{\psi =0}^{\prime }}=\beta \gamma ϵ$. We found the greatest agreement at low ionic strengths (see dashed-dot line for C_KCl=1 mM), for which the spatial decay in the electrostatic potential energy, ϵ(r), and diffuser probability density, C_AMP(r), were mostly linear (see Fig. S7). Given this correspondence for modest electrostatic interaction energies, these data suggest a simple correction to the MG bound of the form.

$$D_{\text{eff}}=\frac{\varphi }{2-\varphi }{e}^{\beta \gamma ϵ}$$ (8)

where γ may be determined by a linear fit to the data in Fig. 8b. This expression is unlikely to hold for strongly attractive interactions (−qψ₀ ≫ 2 – 3 k_BT), which would result in adsorption of the diffuser and a corresponding reduction in diffusion rates.⁷ This topic is further discussed in Sec. Limitations.

In Fig. 8a we compared the HSE model (red square) with continuum simulations (linearized and non-linearized PB equations) for different surface potentials when λ = 7.85 Å, which corresponds to a 150 mM background electrolyte concentration and ϕ = 0.78. Strong agreement for the predictions from the HSE model and the time-dependent models are observed. We additionally present in Fig. 8a effective diffusion rates that calculated using the HSE model for electro-neutral (nearly equal numbers of positively- and negatively-charged crowders) crowder distributions as representations of a typical crowded cytoplasm. For these calculations, we assumed alternating positive/negative crowders (diamond) as well as randomly-charged (circle) or randomly-charged and randomly-distributed crowders (triangle symbols). For alternating positive and negatively charged crowders, D_eff appears to modestly increase (approximately 5%) for increasing ∣ϵ∣. This may arise due to the stronger influence of attractive interactions on diffusion relative to repulsive interactions of the same amplitude. This effect, however, is largely lost when the crowder positions or distribution of positive versus negative crowders are randomized; for these cases, the D_eff coefficients are nearly uniform for interaction strengths of ∣ϵ∣ ≤ 1k_BT and comparable to predictions for an uncharged diffuser.

To understand the dependence of D_eff on the bulk ionic strength for uniformly-charged crowders illustrated in the previous section, we present HSE simulations for crowders in electrolyte solutions ranging from dilute to concentrated. For weak electrostatic interactions occurring predominantly for thin DLs (i.e., κH ≫ 10 ), D_eff approaches the neutral diffuser limit (star symbol). As κH → 0, the electrostatic interactions between crowders and the diffusion medium are shielded to a lesser extent; as a result, the hindrance of diffusion due to repulsive crowder/diffuser interactions is intensified, particularly as the accessible volume fraction is reduced. Similarly, the enhancement of diffusion due to attractive interactions is more apparent at lower κH values. These trends are consistent with the ionic strength dependence of diffusion rates reported in experiment.^71,72 In sharp contrast, for the randomized ensembles of negatively- and positively-charged crowders introduced in the previous section, D_eff is almost entirely independent of κH, except at very low κH where minor enhancement is observed. Together, these data suggest that for randomly distributed cytoplasmic crowders, the excluded volume effect overwhelmingly determines the effective diffusion rate of small diffusing molecules; however, for regions with a disproportionate share of electropositive or electronegative crowders, as might be expected for compacted DNA in the nucleus,⁷⁴ effective diffusion rates may significantly deviate from estimates based on excluded volume alone.

For the latter scenario, we propose a D_eff relationship based on an effective crowder radius determined by the Debye length, λ. For this approximation, we use the functional form of Eq. 7, but assume ϕ is dependent on λ as given by

$$\varphi =\left(1-\frac{\pi (R\pm \gamma {\lambda }_D{)}^{2}N}{L^2}\right)$$ (9)

where +γ and −γ are corrections used for repulsive and attractive interactions, respectively. In contrast to Eq. 8, which is most accurate at low ionic strength (e.g. long Debye lengths), this approximation performs well at moderate to high ionic strengths as shown in Fig. 9 a and b. The γ parameter scales approximately linearly with surface potential independent of accessible volume fraction as illustrated in Fig. S9. The revised analytical bound (line), Eq. 9, agrees well with the diffusion coefficient calculated using HSE for moderate to high κH when γ is 0.45 and 0.66 for repulsive and attractive interactions, respectively and it is independent of the accessible volume fraction. The small deviation at ϕ=0.717 when κH < 2.5 can be attributed to strong DL overlap, for which Eq. 8 is a more appropriate bound.

Figure 9:

Normalized effective diffusion coefficient (via HSE) for a) repulsive and b) attractive electrostatic interactions as a function of κH and varying crowder surface distances (H). Results for H=2.57, 5.2, 9, or 14 nm are denoted with crosses, circles, diamond, triangles, and squares, respectively. Evaluation of MG bound for each volume fraction is represented by a pink star. Solid lines represent empirical fits based on ‘effective’ volume fractions from Eq. 9 (see Fig. S9)

Limitations

Our HSE simulations of small molecule diffusion through matrices of weakly-charged cytosolic proteins offer insight into the influence of protein shape, size, charge, and density on transport within the cell. The fundamental assumption for homogenization theory is that there exists an underlying periodic structure. In biological media, this assumption can only be satisfied approximately, as the distribution of proteins is not strictly periodic. Nevertheless, it is apparent from this study and others^45,75 that homogenization-based predictions of effective diffusion tensors are in excellent agreement with experimental measurements. We attribute this success to the fact that the position of each inclusion does not strongly deviate from the ‘typical’ unit cell, or similarly, the statistical distribution within a given unit cell is consistent across all cells. In fact, this is the basis for a homogenization scheme valid for non-periodic materials,⁷⁶ wherein it is assumed that periodic spherical inclusions are remapped to non-periodic positions that remain strictly confined within the unit cell. Additionally, for examination of systems with periodicity-breaking defects, such as the presence of intracellular organelles in a cytoplasmic protein lattice, aperiodic correctors for localized perturbations may be appropriate.^77,78

There are a number of additional model improvements that may be considered, particularly for describing highly-charged systems. For the crowder surface potentials considered in this study (∣ψ∣ ≤ 25 mV), the linearized PB was sufficient to describe the diffuse layer about charged crowders as well as their overlapping DLs. For more highly-charged diffusers or crowders (∣ψ∣ ≫ 25. mV), numerical solution of the nonlinear PB will be necessary to adequately model the electric potential.⁷⁹ More importantly, stronger attractive electrostatic interactions will introduce several complications that did not arise in our study. As illustrated by the Boltzmann equation, ρ = ρ_bulkexp(−βqψ), attractive interaction energies as low as 3 kcal/mol are sufficient to increase the local diffuser concentration by two orders of magnitude relative to bulk. For the 1 mM AMP concentration assumed in our time-dependent modeling, the local AMP concentration near crowders could approach that of the bulk KCl and would likely significantly enhance electrostatic shielding. This contribution would mandate including both the diffuser concentration and bulk electrolyte (KCl) in the determination of the electrostatic potential; Poisson-Nernst-Planck (PNP) approaches,⁸⁰ whereby the diffusion and PB equations are solved iteratively and concurrently, are routinely used to describe such phenomena. Here, protocols to homogenize the PNP would be a natural extension of our current work.⁴⁸ It may also be advantageous to consider the finite sizes of electrolyte and diffuser particles, which are well-known to attenuate apparent concentrations of counter-ions along charged surfaces compared to theories that assume point charge electrolytes.^81-83 Along these lines, significant surface interactions would likely give rise to adsorption phenomena that may require the consideration of adsorption isotherms when defining the model boundary conditions.⁸⁴ Consideration of additional surface interactions, such as hydrophobic interactions and colloidal aggregation forces, may be advantageous for larger diffusing species, and in many cases could be accommodated by augmenting the qψ term of Eq. 2 with Lennard-Jones or DLVO potentials,⁷⁹ respectively.

We assumed that the diffuser was substantially smaller than the crowders, which justified our use of immobile protein lattices. Inherent in this assumption is that small molecules diffuse much faster than the crowders and thereby rapidly achieve a local steady state before significant displacement of the crowders. A recent Brownian dynamics study from Putzel et al.⁷ examining particle diffusion among larger crowders was also based on this assumption; however, in their Supplement, they also reported effective diffusion rates for systems comprised either entirely or partially of mobile crowders that were 20% to 40% higher than the fixed system. Hence, we anticipate fairly modest increases in diffusivity under the assumption of mobile crowders. However, for larger diffusers than the small biomolecules considered here, hydrodynamic effects and sub-diffusive behavior would likely demand a mobile crowder treatment.^24,25,85

Conclusions

We investigated the effective diffusion rates for small, charged biomolecules within crowded, sub-micron-scale cytoplasm domains as functions of crowder shape, size, distribution, electrostatic interaction energies and ionic strength. Our HSE approach recapitulated observations that neutral diffusers exhibit hindered diffusion as the accessible volume fraction is decreased due to crowding. However, our studies further indicate that in absence of electrostatic interactions, crowder shape, size and distribution variation have minor effects on the effective diffusion coefficient at physiological free volume fractions (ϕ ≈ 0.8). Our method also demonstrates that electrostatic interactions between crowders and diffusers substantially alter D_effs in manners highly-dependent on whether interactions are attractive versus repulsive, the surface potential amplitude, and solvent ionic strength. Generally, media presenting exclusively repulsive interactions tend to hinder diffusion, whereas enhanced diffusion is observed for weakly attractive interactions. In the more likely crowded configuration consisting of randomly-distributed and randomly charged crowders, our homogenization results indicate that typical intracellular accessible volume fractions and crowder interaction strengths are sufficient to reduce effective diffusion rates within an order of magnitude, which is in line with conventional estimates.³ Surprisingly, partitioning of like-charged crowders into sub-cellular domains can strongly perturb diffusion in localized regions of the cell. This co-dependence of D_eff on ϕ and electrostatic interactions can be empirically modeled by slight modifications of the MG bound for neutral crowders (see Eq. 8 and Eq. 9). In our opinion, this raises the possibility that nature may exploit such perturbations to control the efficiency and kinetics of reactions in small, sub-cellular compartments, as is postulated for intracellular nucleotide-, Na⁺- and Ca²⁺-dependent ‘nanodomain’ signaling.^62,86,87 Moreover, given the strong agreement between D_eff estimates from the time-dependent and homogenized electro-diffusion equations, the adaptability of the homogenized Smoluchowski equation to arbitrary potentials of mean force (see Eq. 2) could facilitate examination of how additional prominent physical phenomena, including adsorption and buffering,⁴¹ shape intracellular communication in crowded, biological systems.